Department of Statistics, University of Manitoba, Winnipeg, Canada

Abstract

The ranking method used for testing the equivalence of two distributions has been studied for decades and is widely adopted for its simplicity. However, due to the complexity of calculations, the power of the test is either estimated by a normal approximation or found when an appropriate alternative is given. Here, via the Finite Markov chain imbedding technique, we are able to establish the marginal and joint distributions of the rank statistics considering the shift and scale parameters, respectively and simultaneously, under two different continuous distribution functions. Furthermore, the procedures of distribution equivalence tests and their power functions are discussed. Numerical results of a joint distribution of rank statistics under the standard normal distribution and the powers for a sequence of alternative normal distributions with means from −20 to 20 and standard deviations from 1 to 9 and their reciprocal are presented. In addition, we discuss the powers of the rank statistics under the Lehmann alternatives.

2010 Mathematics Subject Classification

Primary 62G07; Secondary 62G10

1 Introduction

Suppose that on the basis of observations _{1},…,_{
m
};_{1},…,_{
n
} from the cumulative distribution functions

which is known as the shift alternative and, for some

Wilcoxon (1945) proposed the ranking method for testing the significance of the difference of the two populations means, also known as the Wilcoxon rank-sum test, and defined a statistic _{
Y
}, as the sum of the ranks of the ^{′}
^{′}
^{′}

Mann and Whitney (1947) introduced an elaboration of the ranking test, proposed the statistic _{
X
} is

as

and

with

where ^{′} and ^{′} are independently distributed, ^{′} with the distribution ^{′} with the distribution

where

Over the years, there have been studies on finding the exact or approximate power for the rank-sum test. By choosing an appropriate alternative distribution function, Shieh et al. (2006) derived the exact power for the uniform, normal, double exponential and exponential shift models. Rosner and Glynn (2009) discussed power against the family of alternatives of the form

where the underlying distributions _{
X
} and _{
Y
} are normal. Collings and Hamilton (1988) presented a bootstrap method to find the empirical distribution functions in order to approximate the power against the shift alternative. Lehmann (1953) derived the power function as

where _{
j
} is the rank of _{
j
} in the combined samples for the alternative hypothesis of

where

As the rank-sum test is widely adopted for testing the center differences of two distributions, it is natural to study the efficiency of a rank-sum test for variability (Ansari and Bradley 1960). For decades, studies have focused on proposing new definitions of the rank statistic and using the methods of Chernoff and Savage to show the relative efficiency of the proposed statistic to the F-test, see for example Mood (1954), Siegel and Tukey (1960), Ansari and Bradley (1960), and Klotz (1962). Ansari and Bradley (1960) mentioned that if the means of the

Our approach aims at releasing some of the conditions for finding the distribution of the proposed rank statistic. We systematically imbed the random vector **
U
**

The main contributions of this paper are as follows. In Section 2.1, we introduce the procedures of deriving the distribution of the rank statistic considering the shift parameter and its power function by using FMCI. The procedures are general and can be applied to either two identical distribution functions of interest or two different continuous density functions. In Section 2.2, we address the steps for finding the distribution of the rank statistic considering the scale parameter and its power function. In Section 2.3, we retrieve the joint distribution of the rank statistics considering the location and scale parameters simultaneously as well as its power function. Numerical results of a joint distribution and some powers of the rank statistics against shift parameter and scale parameter, individually and simultaneously, are presented in Section 3. We also discuss the powers of the rank statistics under the Lehmann alternatives. We end this paper with a short conclusion in Section 4.

2 Methods

2.1 Distributions of the rank statistic in the shift case

Let {_{1},…,_{
m
}} and {_{1},…,_{
n
}} be two independent samples from the continuous cumulative density distributions **
x
**={

for _{[0]}=−_{[m+1]}=

Given

where _{
i
}(^{′}
_{[i−1]},_{[i]}) among _{1},…,_{
t
}. For each **
u
**

**Theorem 1. **_{
l
} _{
Y
},

The rank statistic _{
Y
}, sum of the ranks of

The first summation of the first term in Equation (5) can be interpreted as the number of _{[i]} which is

It is then easy to see that

Next, we demonstrate that for two random samples from the same population, the distribution of the random vector **
U
**

**Theorem 2. ****
U
**

^{′}

and, when **
U
**

where _{[0]}=−_{[m+1]}=

Using variable transformation, it is clear to see that the random variables _{[1]}),…,_{[m]}) have a Dirichlet distribution with parameters _{1}(_{2}(_{
m+1}(

which is independent of the distribution function.

This is the reason that the distribution of the rank statistic **
U
**

Let _{
t
},

possible states, _{
n
}={0,1,…,_{
t
}:_{
n
}} be a non-homogeneous Markov chain on the state space _{
t
}. As a transition probability matrix **
M
**

where

and _{
i
} is defined in Equation (3).

**Theorem 3. **
_{
l
}(**
U
**

**
u
**

**
u
**

where **
U
**

Then, the Law of Large Numbers is used to determine the probability of **
U
**

where **
X
**

To test

for some

where

Note that the alternative hypothesis is subject to the purpose of the test. This simply needs to be slightly modified if a one-sided test is adopted.

2.2 Distributions of the rank statistic in the scale case

We studied the distribution and the power function of the rank statistic _{
l
} considering a shift in location. Now, the distribution and the power function of the rank statistic considering the scale parameter will be addressed. For this purpose, we consider ^{−1}) and state the null and alternative hypotheses as

To do so, we begin with the procedure of finding the distribution of the rank statistic, denoted _{
s
}, considering the scale parameter through the random vector **U**
_{
n
}. The array of ranks are given by

if

if ^{′}
_{
s
}, with respect to

where _{
i
}(_{[i−1]},_{[i]}). Let ^{′}
^{′}
_{[i]},_{[i+1]}) which will then break **U**
_{
n
} into two parts _{[i]}, then

is a 1×

is a 1×(

and

The third possible case is, if _{[i]} is the smallest number larger than

and

The last possibility is, if

and

Let ^{−} be the length of the vector ^{+} be the length of the vector

**Theorem 4. **
_{
s
}(**U**
_{
n
}|**
X
**)

**U**
_{
n
}, **
ξ
**(=

**U**
_{
n
} in the state space _{
n
}, we have a corresponding

The smallest possible value of _{
s
}(**U**
_{
n
}) is

and the largest possible value is

In accordance with Equation (11), we use the possible value of _{
s
} as a rule of the partition. The rest of the proof follows along the same line as that of Theorem 3, and here, is omitted.

Similarly, we apply the LLN to conclude that

which establishes the distribution of _{
s
}.

Through FMCI we, again, successfully retrieved the distribution of _{
s
} under selected alternative distributions, for which the procedures are similar to those in the previous section. In addition, it is quite intuitive to approximate the power function by

where

2.3 Joint distributions of the rank statistics in the shift and scale case

We have derived the marginal distributions of _{
l
} and _{
s
} in terms of **
U
**

**Theorem 5.** (_{
l
}(**
U
**

**
u
**

**
u
**

The joint distribution of the ranks considering both the location and scale parameters which can be determined through our algorithm is yet to be studied in the literature. Our result allows us to test the homogeneity of the distribution functions ^{−1}). We state the hypotheses as follows

Also we are able to identify a proper critical region under the null hypothesis and discuss its power when

where _{1l
}, _{2l
}, _{1s
} and _{2s
} are the critical values such that

or an elliptic critical region

for some positive constants

According to the above defined rejection region, the power of the test can be found as

or

Note that unless having a conjecture about the values of

3 Numerical results and discussion

3.1 A joint distribution of _{
l
} and _{
s
}

Let {_{1},…,_{5}}∼_{1},…,_{7}}∼_{
l
} and _{
s
} under the null hypothesis of _{
l
} and _{
s
} can be easily established from their joint distribution. Figure _{
l
} and _{
s
} are dependent. We construct two critical regions as shown in Figure _{0.1738} and outside the red shadow is the elliptic one

Joint distribution of _{l} and _{s} in the case where

**Joint distribution of**
**
R
**

Critical Regions at size 17.38% for _{l} and _{s} for

**Critical Regions at size 17.38% for**
**
R
**

3.2 Powers for a joint test using _{
l
} and _{
s
}

The alternative of interest is stated in the preceding section (see Equation (14)). The power functions of the test statistics _{
l
} and _{
s
} for a sequence of normally distributed populations with

Power functions of _{l} and _{s} for _{α}.

**Power functions of**
**
R
**

Power functions of _{l} and _{s} for

**Power functions of**
**
R
**

Power comparisons of the joint test _{l} and _{s} for

**Power comparisons of the joint test**
**
R
**

Next, we consider the problem of determining an optimum rank test. To conduct a test of distributions equivalency, we can use either _{
l
} or _{
s
} as the test statistic. As mentioned earlier, the marginal distribution _{
l
} or _{
s
} can be easily established from their joint distribution. Figures _{
l
} and _{
s
} at the level of significance 17.38%, respectively. Figure _{
l
} or _{
s
} alone for distributions equivalence tests. A joint test for distributions equivalency would like a better option under most circumstances.

Power functions of _{l} given

**Power functions of**
**
R
**

Power functions of _{s} given

**Power functions of**
**
R
**

3.3 Lehmann alternatives

Consider the one-sided alternative _{
o
}:_{
a
}:^{
k
}= ^{
k
} is the cumulative distribution of max1≤_{
i
}) when _{
i
}∼

Therefore, the larger the _{
l
} is, the stronger the evidence against the null hypothesis will be. For the variation of the distribution per se, the codomain of the density function is compressed to larger numbers; therefore, in most cases, we have _{
k
}) < _{
s
} is large. For example, given ^{
k
}, it is easy to see

and

for all _{
l
} and _{
s
} in order to define critical regions for _{
l
} and _{
s
} individually and simultaneously. Due to the properties of the mean and variance of the alternative distribution, as shown in Equations (17), (18) and (19), we are cautious to define the critical regions. Table _{
l
} and _{
s
} for the equality of distributions is best suited in comparison with tests considering only one of the rank statistics.

**
m
**

**
m
**

**
m
**

**F**

**Test**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

**
β
**

Note: A sectorial critical region is chosen for a simultaneous testing.

_{
l
}

.090

.411

.647

.900

.096

.496

.761

.967

.099

.591

.845

.984

_{
s
}

.080

.152

.193

.218

.076

.137

.149

.123

.100

.236

.370

.638

_{
l
}&_{
s
}

.100

.452

.699

.934

.100

.531

.799

.981

.100

.622

.878

.992

_{
l
}

0.090

.412

.639

.897

0.096

.493

.756

.965

0.099

.574

.841

.987

_{
s
}

0.080

.150

.197

.217

0.076

.137

.152

.121

0.100

.234

.367

.634

_{
l
}&_{
s
}

0.100

.453

.696

.932

0.100

.528

.798

.980

0.100

.606

.874

.993

_{
l
}

0.090

.411

.650

.899

0.096

.490

.764

.967

0.099

.579

.841

.987

_{
s
}

0.080

.149

.195

.217

0.076

.140

.152

.122

0.100

.232

.376

.641

_{
l
}&_{
s
}

0.100

.451

.702

.933

0.100

.525

.805

.982

0.100

.607

.875

.993

4 Conclusion

Our proposed algorithm provides a solution for finding the power of distribution equivalence tests considering the shift and scale parameters, respectively and simultaneously. Numerical studies show that a joint test should be adopted for the test homogeneity of distributions as well as under Lehmann alternatives. Also an elliptic critical region is a better choice rather than a rectangular one for a joint test. In practice, it is reasonable to have neither the normality assumption nor equal mean/variance of the interested distributions. However, our algorithm highly depends on the technology equipments as the possible states in _{
n
} grow rapidly when the sample sizes increase. Therefore, we can, so far, only target small sample sizes in our work.

Competing interests

The author declares that she has no competing interests.

Acknowledgments

The author would like to thank James C. Fu and anonymous referee whose comments led to significant improvements of this manuscript.